Transport phenomena-Mass Transfer 31-Jul-2016

Transport Phenomena Mass Transfer

(1 Credit Hour)

Dr. Muhammad Rashid Usman Associate professor

Institute of Chemical Engineering and Technology

University of the Punjab, Lahore.

Jul-2016

μ α k ν DAB Ui Uo UD

hi ho Pr f Gr Re Le

Nu Sh Pe Sc kc Kc d

Δ ρ Σ Π ∂ ∫

2

The Text Book

Bird, R.B. Stewart, W.E. and Lightfoot, E.N. (2002). Transport

Phenomena. 2nd ed. John Wiley & Sons, Inc. Singapore.

Please read

through.

http://www.google.com.pk/url?sa=i&rct=j&q=transport+phenomena+by+bird&source=images&cd=&cad=rja&docid=7tehW-KOHJ6wnM&tbnid=XLsTMM3_-9N4pM:&ved=0CAUQjRw&url=http%3A%2F%2Fwww.goodreads.com%2Fbook%2Fshow%2F842230.Transport_Phenomena&ei=lyXTUdrNMcaOOOz5gIgK&bvm=bv.48705608,d.bGE&psig=AFQjCNEZ1zZmqJByJL_kb6P9Ddy3lGL9Eg&ust=1372878586174902

3

Transfer processes

For a transfer or rate process

Conductance is a transport property.

force drivingquantity a of Rate

force drivingArearesistance

quantity a of Rate

1

force drivingAreaonductancec quantity a of Rate

Compare the above equations with Ohm’s law of electrical

conductance

quantity the of flow the for reaaquantity a of Rate

force drivingonductancec quantity a of luxF

4

Transfer processes

quantity the of flow for area

quantity the of ratequantity a of Flux

time in change

quanity the in changequantity a of ateR

distance in change

quanity the in changequantity a of radientG

5

Transfer processes

In chemical engineering, we study three transfer

processes (rate processes), namely

•Momentum transfer or Fluid flow

•Heat transfer

•Mass transfer

The study of these three processes is called as

transport phenomena.

6

Transfer processes

Transfer processes are either:

• Molecular (rate of transfer is only a function

of molecular activity), or

• Convective (rate of transfer is mainly due to

fluid motion or convective currents)

Unlike momentum and mass transfer processes,

heat transfer has an added mode of transfer

called as radiation heat transfer.

7

Summary of transfer processes

Rate Process Driving force

Conductance

(Transport property)

Law for molecular

transfer

Momentum

transfer Velocity gradient

Viscosity

(Kinematic viscosity)

Newton’s law of

viscosity

Heat transfer Temperature

gradient

Thermal conductivity

(Thermal diffusivity) Fourier’s law

Mass transfer

Concentration

gradient (chemical

potential gradient)

Mass diffusivity Fick’s law

8

Molecular rate laws

Newton’s law of

viscosity (modified)

Fourier’s law of

conduction heat transfer

(modified)

Fick’s law of molecular

diffusion

9

Molecular rate laws

Each transport diffusivity has the SI unit of m2/s.

are concentration terms and are

momentum, heat, and mass (molar)

concentrations having SI units of

(kg·m·s‒1)/m3, J/m3, and mol/m3,

respectively.

transport properties.

transport diffusivities.

10

Ratios of transport diffusivities

Prandtl number is the ratio of molecular diffusivity of

momentum (kinematic viscosity) to molecular

diffusivity of heat (thermal diffusivity).

Schmidt number is the ratio of molecular diffusivity of

momentum (kinematic viscosity) to molecular

diffusivity of mass.

Lewis number is the ratio of molecular diffusivity

of heat (thermal diffusivity) to molecular

diffusivity of mass.

11

Ratios of transport diffusivities

12

Rapid and brief introduction to mass transfer

Mass transfer is basically of two types

•Molecular mass transfer

•Convective mass transfer

Mass transfer can be within a single phase

(homogeneous) or between phases. In the latter

case it is called as “interphase mass transfer”.

13


Fick’s Law of mass transfer

A more general driving force is based on chemical

potential gradient (μc)

dz

dcDJ A

AAz

dz

d

RT

DcJ cA

AAz

14


Where,

μo is the chemical potential at the standard state.

Apart from difference in concentration, a chemical

potential gradient can also be obtained by

temperature difference (thermal diffusion or Soret

effect), pressure difference, differences in gravity

forces, magnetic forces, etc. See Ref. 5, Chapter 24.

Ao

c cRT ln

15


16

Rapid and brief introduction to mass transfer [2]

17


)( zAzAAz uucJ

For a binary system (A and B) with a constant

average velocity in the z-direction, the molar flux

in the z-direction relative to the molar-average

velocity may be expressed by:

18

Turbulent mass diffusivities

19


20


21

Convective mass transfer

22


23


24


25

Convective mass transfer correlations

26

Interphase mass transfer: Two film theory

pA

pAi

CAi

CA

Main body

of gas

Main body

of liquid

Liquid film

Gas film

CAi

pAi

Interface (assume

equilibrium at interface and

use Henry’s law with

constant value of Henry’s

constant)

27

Two-film theory: Overall mass transfer coefficient

28

Diffusion coefficient or mass diffusivity

The proportionality coefficient for the Fick’s law for

molecular transfer is mass diffusivity or diffusion

coefficient.

Like viscosity and thermal conductivity, diffusion

coefficient is also measured under molecular transport

conditions and not in the presence of eddies (turbulent

flow).

The SI units of mass diffusivity are m2/s.

1 m2/s = 3.875×104 ft2/h

1 cm2/s = 10‒4 m2/s = 3.875 ft2/h

1 m2/h = 10.764 ft2/h

1 centistokes = 10‒2 cm2/s

29


Why do we use a small diameter capillary for the

measurement of viscosity of a liquid?

Bring to mind Ostwald’s viscometer.

30

Ranges of mass diffusivities

Diffusion coefficients are generally highest for

gases, lower for liquids, and lowest for solids.

The ranges are as follow [4]:

For gases: 5×10‒6 to 1×10‒5 m2/s

For liquids: 1×10‒10 to 1×10‒9 m2/s

For solids: 1×10‒14 to 1×10‒10 m2/s

31


Diffusion coefficients may be obtained by doing

experiments in the laboratory.

Diffusion coefficients may be obtained from

experimentally measured values by various

researchers in the field. Many of these data is

collected in handbooks, etc.

When experimental values are not found, diffusion

coefficients may be estimated using theoretical,

semi-empirical (semi-theoretical), or empirical

correlations.

32

Experimental diffusion coefficients [6]

33

Experimental diffusion coefficients: References

Lide, D.R. 2007. CRC Handbook of chemistry and

physics. 87th ed. CRC Press.

Poling, B.E.; Prausnitz, J.H.; O’Connell, J.P. 2000.

The properties of gases and liquids. 5th ed. McGraw-

Hill.

Green. D.W.; Perry, R.H. 2008. Perry’s chemical

engineers’ handbook. 8th ed. McGraw-Hill.

Please populate the list.

34

Diffusion coefficient for gases at low pressure

For gases at low pressure (low density gases), the

diffusion coefficient

o decreases with pressure

o increases with temperature, [4], and

o virtually remains constant with gas composition.

35

Diffusion coefficient for gases at low pressure: Estimation

Based on kinetic theory and corresponding states

principle the following equation [1] can be used for

obtaining the mass diffusivity of gases at low pressures.

Here, DAB is in cm2/s, T in K, and p in atm. See Table below.

36

The equation fits the experimental data well at atmospheric

pressure within an average error of 6‒8%.

Situation a b

For non-polar gas pairs (excluding He and H2)

2.745×10‒4 1.823

H2O and a nonpolar gas 3.640×10‒4 2.334


37


Example 17.2-1 [1]:

Estimate DAB for the system CO-CO2 at 296.1 K and 1

atm total pressure. Use the equation just described. A is

CO and B is CO2. The molecular weight and critical

properties of the species are given as below:

38

Homework problem

Estimate DAB for the systems CO2-N2O, CO2-N2, CO2-

O2 , N2-O2, and H2-N2 at 273.2 and 1 atm and compare

the results with the experimental values given in Table

17.1-1 [1]. Also for H2O-N2 at 308 K and 1 atm and

compare similarly.

39


Based on the Chapman-Enskog theory, the following

equation (sometimes called as Hirschfelder equation) using

ideal gas law for the determination of total concentration “c”

may be written:

Here, DAB is in cm2/s, σAB is Å, and p in atm.

The mass diffusivity DAB for binary mixtures of nonpolar

gases is predictable within about 5% by kinetic theory.

40


The parameters σAB and εAB could, in principle, be determined

directly from accurate measurement of DAB over a wide range of

temperatures. Suitable data are not yet available for many gas

pairs, one may have to resort to using some other measurable

property, such as the viscosity of a binary mixture of A and B. In

the event that there are no such data, then we can estimate them

from the following combining rules for non-polar gas pairs.

Use of these combining rules enables us to predict values of DAB

within about 6% by use of viscosity data on the pure species A

and B, or within about 10% if the Lennard-Jones parameters for A

and B are estimated from boiling point data as discussed in

Chapter 1 of the text [1].

41


The Lennard-Jones parameters can be obtained from the

Appendix of the text book [1] and in cases when not known can

be estimated as below:

The boiling conditions are normal conditions, i.e., at 1 atm.

42

Lennard-Jones potential [7]

43


The collision integral can be obtained using dimensionless

temperature from the Appendix of the text book [1]. To

avoid interpolation, a graphical relationship for collision integral

is given in the next slide.

A mathematical relationship by Neufeld is recommended for

computational (spreadsheet) work [5].

44

Diffusion coefficient for gases at low pressure: Estimation [4]

45


The effect of composition (concentration) is not

accommodated in the above equation. The effect of

concentration in real gases is usually negligible and

easily ignored, especially at low pressures. For high

pressure gases, some effects, though small, are

observed.

46


Example 17.3-1 [1]:

Predict the value of DAB for the system CO-CO2 at

296.1 K and 1 atm total pressure.

Use the equation based on Chapman-Enskog theory as

discussed above. The values of Lennard-Jones Potential

and others may be found in Appendix E of the text [1].

47


48


The above equation (Hirschfelder equation) is used to

extrapolate the experimental data. Using the above

equation, the diffusion coefficient can be predicted at

any temperature and any pressure below 25 atm [4]

using the following equation.

2

49

Fuller’s equation

Fuller’s equation is an empirical equation that is also

used at low pressures. The equation may be used for

nopolar gas mixtures and polar-nonpolar mixtures [2].

The equation is less reliable than that of Hirschfelder

equation described above.

Use T in K, p in atm, υ from the table given on the next

slide. DAB will be in cm2/s.

50

Diffusion volumes to be used with Fuller’s equation [4]

51

Use of Fuller’s equation

Example problem [2]:

Normal butanol (A) is diffusing through air (B) at 1 atm

abs. Using the Fuller’s equation, estimate the diffusivity

DAB for the following conditions:

a) For 0 °C

b) 25.9 °C

c) 0 °C and 2.0 atm abs.

52

Homework Problems

1. Calculate the diffusion coefficient of methane in air

at 20 oC and 1 atm. Use Fuller’s equation.

2. Using Fuller’s equation, calculate the diffusion

coefficient of C2H5OH in air at 298 K and 1 atm.

53

Diffusion coefficient for multicomponent gas mixtures

Blanc’s law:

May be good to ternary systems in which

component i has very low concentration [5].

See Poling et al. [5] for additional discussion.

54

Diffusion coefficient for gases at high density

The behavior of gases at high pressures (dense

gases) and of liquids is not well understood and

difficult to interpret to a mathematical

evaluation. The following approaches can be

used for estimating the mass diffusivity of gases

at high pressures.

55

Diffusion coefficient for gases at high density: Takahashi method [5]

Here, Tr and pr are pseudoreduced temperature and

pseudoreduced pressure, respectively. p is in bar and

DAB is in cm2/s. + sign suggests value at low pressure.

56

Diffusion coefficient for gases at high density: Takahashi method [5]

Tr

57

Diffusion coefficient for gas mixtures [1]

58

Diffusion coefficient for gas mixtures [1]

The value of can be obtained from the graph above

by knowing reduced temperature and reduced pressure.

Where, DAB is in cm2/s, T in K, and pc in atm.

59

Diffusion coefficient for gas mixtures

The above method is quite good for gases at low

pressures. At high pressures, due to the lack of large

number of experimental data, a comparison is difficult

to make. The method is therefore suggested only for

gases at low pressures.

The value of cDAB rather than DAB is because

cDAB is usually found in mass transfer

expressions and the dependence of pressure and

temperature is simpler [1].

60

Diffusion coefficient for gas mixtures at high density

Example 17.2-3 [1]:

Estimate cDAB for a mixture of 80 mol% CH4 and 20

mol% C2H6 at 136 atm and 313 K. It is known that, at 1

atm and 293 K, the molar density is c = 4.17 × 10‒5

gmol/cm3 and DAB 0.163 cm2/s.

61

Problem 17 A.1 [1]

62

Diffusion coefficient for liquid mixtures

Due to the not well defined theory for diffusion in

liquids, empirical methods are required for the

estimation of diffusion coefficients in liquids.

The following Wilke-Chang equation may be used for

small concentration of A in B [1]:

63


In the above equation, Ṽ is molar volume of solute A

in cm3/mol as liquid at its boiling point, μ is viscosity of

solution in cP, ψB is an association parameter for the

solvent, and T is absolute temperature in K.

Recommended values of ψB are: 2.6 for water; 1.9 for

methanol; 1.5 for ethanol; 1.0 for benzene, ether,

heptane, and other unassociated solvents.

The equation is good for dilute solutions of

nondissociating solutes. For such solutions, it is usually

good within ±10%.

64

Selected molar volumes at normal boiling points [4]

65

Atomic volumes for calculating molar volumes and corrections [4]

66


When molar volumes are not available, one may use the

following equation after Tyn and Calus:

Vc is critical volume in cm3/gmol.

67


Example 17.4-1 [1]:

Estimate DAB for a dilute solution of TNT (2,4,6-

trinitrotoluene) in benzene at 15 °C. Take TNT as A and

toluene as B component. Molar volume of TNT is 140

cm3/mol.

Hint: As dilute solution, use viscosity of benzene

instead of solution.

Example 5 [4, 418]:

Estimate the diffusion coefficient of ethanol in a dilute

solution of water at 10 °C.

68

Homework problem

Example 6.3-2 [2]:

Predict the diffusion coefficient of acetone

(CH3COCH3) in water at 25 °C and 50 °C using the

Wilke-Chang equation. The experimental value is 1.28

× 10‒9 m2/s at 25 °C.

69

Further reading

Poling, B.E.; Prausnitz,

J.H.; O’Connell, J.P.

2000. The properties of

gases and liquids. 5th ed.

McGraw-Hill.

Excellent reference for

the estimation of mass

diffusivities. See Chapter

11 of the book.

70

Homework problems

Questions for discussions [1, pp. 538–539]:

1, 3, 4, and 7.

Problems [1, pp. 539–541]:

17A.4, 17A.5, 17A.6, 17A.9, and 17A.10.

71

Shell mass balance

Why do we need shell (small

control element ) mass balance?

72

Shell mass balance

We need, because, we desire to know what is happening

inside. In other words, we want to know the differential

or point to point molar flux and concentration

distributions with in a system, i.e., we will represent the

system in terms of differential equations. Macro or bulk

balances such as that applied in material balance

calculations only give input and output information and

do not tell what happens inside the system.

73

Steps involved in shell mass balance

o Select a suitable coordinate system (rectangular,

cylindrical, or spherical) to represent the given problem.

o Accordingly, select a suitable small control element or

thin shell of finite dimensions in the given system.

o Define the appropriate assumptions (steady-state, etc.)

and apply mass balance over the shell geometry.

o Let the shell dimensions approach zero (differential

dimensions) and using the definition of derivation, obtain

the differential equations that describe the system. If only

one direction is considered there will be one first order

differential equation for a steady-state process. The

solution of this equation gives the molar flux distribution.

74

Steps involved in shell mass balance

o Insert the definition of molar flux in the above

differential equation, a second order differential

equation will be the result.

o Define the appropriate boundary conditions and solve

the second order differential equation to obtain the

concentration profile and the values of the average flux.

75

Coordinate systems [8]

See Appendix A.6 of the text [1].

76

General mass balance equation

Rewrite the balance in

terms of number of moles

mass of onaccumulati of Ratenconsumptio mass of Rate

generation mass of Rateout mass of Ratein mass of Rate

77

Modeling problem

Diffusion through a stagnant gas film

78

Diffusion through a stagnant gas film [1]

Concentration

distribution

Shell

79


The mass balance (law of conservation of mass) around

the thin shell selected in the column of the gas may be

written for steady-state conditions as:

= 0

80


As no reaction is occurring so the third term on the

previous slide is not needed. The mass balance then can

be written as

Units of are

Where, S is the cross-sectional area of the column and

NAZ is molar flux of A in z direction.

0 zzAzzAz NSNS

s

mol

sm

molm

2

2

AzNS

81


Dividing throughput by and taking , it may

be shown that

We know for combined molecular and convective flux

As gas B is stagnant (non-diffusing), so and the

above equation becomes

0dz

dNAz

z 0z

)( BzAzAA

ABAz NNxdz

dxcDN

0BzN

AzAA

ABAz Nxdz

dxcDN

82


dz

dx

x

cDN A

A

ABAz

1

01

dz

dx

x

cD

dz

d A

A

AB

83

Diffusion through a stagnant gas film: Concentration profile

12

1

1

2

1 1

1

1

1 zz

zz

A

A

A

A

x

x

x

x

12

1

1

2

1

zz

zz

B

B

B

B

x

x

x

x

84

Diffusion through a stagnant gas film: Molar flux

dz

dx

x

cDN A

A

ABAz

1

1

2

12 1

1ln

)( A

AABAz

x

x

zz

cDN

1

2

12

ln)( B

BABAz

x

x

zz

cDN

85


lmB

AAABAz

x

xx

zz

cDN

,

21

12 )(

lmB

AAABAz

p

pp

zzRT

pDN

,

21

12 )(

Using ideal gas law

1

2

21,

1

1ln

A

A

AAlmB

x

x

xxx

1

2

12,

lnB

B

BBlmB

x

x

xxx

or

Where

86

Pseudosteady state diffusion through a stagnant gas film

2)(

20

2

21

,, tt

AAA

lmBLA

AB

zz

txxcM

xD

http://www.consultexim.hu/pdf/cer.pdf

[accessed on: 23-Aug-2015]

http://www.consultexim.hu/pdf/cer.pdf

87

Diffusion through a stagnant gas film: Applications

The results are useful in:

o measuring the mass diffusion coefficient

o studying film models for mass transfer.

88

Diffusion through a stagnant gas film: Defects

In the present model, “it is assumed that there is a sharp transition

from a stagnant film to a well-mixed fluid in which the

concentration gradients are negligible. Although this model is

physically unrealistic, it has nevertheless proven useful as a

simplified picture for correlating mass transfer coefficients” [1]

The present method for measuring gas-phase diffusivities

involves several defects:

“the cooling of the liquid by evaporation

the concentration of nonvolatile impurities at the interface

the climbing of the liquid up the walls of the tube and

curvature of the meniscus” [1]

89

Problem [9]

The water surface in an open cylindrical tank is 25 ft below the

top. Dry air is blown over the top of the tank and the entire

system is maintained at 65 °F and 1 atm. If the air in the tank is

stagnant, determine the diffusion rate of the water. Use diffusivity

as 0.97 ft2/h at 65 °F water in air 1 atm.

[Universal gas constant = 0.73 ft3-atm/lbmol-°R]

Hint: We need vapor pressure of water at 60oF for this problem.

From where one can find the right value of

vapor pressure of water?

90

Problem [9]

At the bottom of a cylindrical container is n-butanol. Pure air is

passed over the open top of the container. The pressure is 1 atm

and the temperature is 70 °F. The diffusivity of air-n-butanol is

8.57×10‒6 m2/s at the given conditions. If the surface of n-butanol

is 6.0 ft below the top f the container, calculate the diffusion rate

of n-butanol. Vapor pressure of n-butanol is 0.009 atm at the

given conditions.

[Universal gas constant = 0.08205 m3-atm/kmol-K]

91

Homework problems

Example 18.2-2 [1]

Problem 18A.1 and 18A.6 [1]

92

Diffusion of A through stagnant gas film in spherical coordinates

A spherical particle A having radius r1 is suspended in a

gas B. The component A diffuses through the gas film

having radius r2. If the gas B is stagnant, i.e., not

diffusing into A, derive an expression for molar flux of

A into B.

Gas B

Particle A

93


Applying shell mass balance and taking limit

when ∆r approaches to zero:

0

2

dr

Nrd Ar

)( BrArAA

ABAr NNxdr

dxcDN

dr

dx

x

cDN A

A

ABAr

1

2221

21

2ArArAr NrNrNr

94


1

2

1

2

12

1 ln)( B

BABAr

x

x

r

r

rr

cDN

1

2

1

2

12

11

1ln

)( A

AABAr

x

x

r

r

rr

cDN

95


Derive concentration profile as shown below for the system just

discussed in the previous slide.

)/1()/1(

)/1()/1(

1

2

1

21

1

1

1

1

1 rr

rr

A

A

A

A

x

x

x

x

96

Diffusion of A through stagnant gas film in cylindrical coordinates

A cylindrical particle of benzoic acid A having

radius r1 is suspended in a gas B. The component

A diffuses through the gas film having radius r2.

If the gas B is stagnant, i.e., not diffusing into A,

derive an expression for molar flux of A into B.

Also work out the concentration profile for the

situation.

97

Diffusion of A through stagnant gas film in cylindrical coordinates: Pls confirm

Applying shell mass balance and taking limit

when ∆r approaches to zero:

0

dr

rNd Ar

2211 ArArAr NrNrrN

1

2

1

2

11

ln

1

1ln

r

r

x

x

cDNrA

A

ABAr

)( BrArAA

ABAr NNxdr

dxcDN

dr

dx

x

cDN A

A

ABAr

1

98

Diffusion of A through Pyrex glass tube

99

Diffusion of A through Pyrex glass tube

)/ln(

)(

121

211

RRR

ccDN AAAB

AR

)/ln(

)/ln(

12

2

21

2

RR

rR

cc

cc

AA

AA

)/ln(

)(

121

211

RRR

xxcDN AAAB

AR

dr

dxcDN A

ABAr

0

d

rNd Ar

100

Separation of helium from gas mixture[4]

101

Homework problems

Problem 18B.7 [1]

Problem 18B.9 [1]

102

Diffusion with a heterogeneous chemical reaction [1]

Consider a catalytic reactor shown on the next slide. At the

catalyst surface the following reaction

occurs instantaneously. Each catalyst particle is surrounded by a

stagnant gas film through which A has to diffuse to reach the

catalyst surface. The component A and B diffuse back out through

the gas film to the main bulk stream. Develop an expression for

the rate of conversion of A to B when the effective gas film

thickness and the main stream concentration xA0 and xB0 are

known. The gas film is isothermal. Hint: as moles of B

are equal to half the moles of A, so the flux of B is half the flux of

A.

BA2

103

Diffusion with a heterogeneous chemical reaction [1]

In the previous case, a very small portion (surface) of

the catalyst is selected for the analysis which may be

regarded as straight surface (being so small) instead of a

curved surface. The curvature effects are negligible.

Imagine Earth, a huge spherical body, but, for a small

portion (area) we do not see any of that curvature.

For the reaction, stoichiometry suggets us that

molar flux of A is two times the molar flux of B, i.e.,

molar flux of A is higher than molar flux of B.

Moreover, the molar flux of A is in opposite direction to

the molar flux of B. Therefore NBz is not equal to 2

times of NAz but the opposite, i.e.,.

104

Diffusion with a heterogeneous chemical reaction

BA2

105


02

11

1ln

2

A

ABAZ

x

cDN

)/(1

02

11

2

11

z

AA xx

106

Diffusion with a heterogeneous chemical reaction [2, 458]

Pure gas A diffuses from point 1 at a partial pressure of 101.32

kPa to point 2 a distance 2.0 mm away. At point 2 it undergoes a

chemical reaction at the catalyst surface and .

Component B diffuses back at steady-state. The total pressure is p

= 101.32 kPa. The temperature is 300 K and DAB = 0.15×10‒4

m2/s.

a) For instantaneous rate of reaction, calculate xA2 and NA.

b) For a slow reaction where k = 5.63×10‒3 m/s, calculate xA2

and NA.

BA 2

Hints:

2

1

1

1ln

A

AABAZ

x

xcDN

107


Pseudo-homogeneous model of a vertical and cylindrical

fixed bed reactor design is to be developed. The feed enters

from the top and flows downwards and leave at the bottom.

Consider isothermal conditions and negligible pressure.

Take A as reactant and apply shell mass balance for A to

develop the differential equation involved in the reactor

system.

a) Consider axial direction only but include axial diffusivity

b) Consider also radial diffusion into Part (a)

c) How can you reduce Part (a) to simple plug flow rate

equation.

d) Describe boundary conditions applied in each case.

108

Steady-state two-dimensional pseudo-homogeneous model for a fixed bed reactor

Shell

Satterfield, C.N. 1991.

Heterogeneous catalysis in

industrial practice, 2nd ed.

Krieger Publishing Co.

http://www.sud-chemie-india.com/prod.htm



balance Mass

:

109

Steady-state two-dimensional pseudo-homogeneous model for a fixed bed reactor

110

Steady-state two-dimensional pseudo-homogeneous model

Continuity equation (constant Dr and Da):

It is two-dimensional second order partial

differential equation.

For the derivations, see class notes and book by Missen, Ref. 10.

111


Boundary conditions:

The simplest boundary conditions are that of Dirichlet,

derivative boundary conditions are called as Neumann

boundary conditions. With possible axial diffusion,

Danckwerts boundary conditions are the correct one.

Though Danckwerts boundary conditions are proposed

earlier by Langmuir, but, they are commonly known

after Danckwerts.

112


113


Continuity equation (negligible axial diffusion,

and constant Dr):

It is two-dimensional parabolic second

order partial differential equation.


114


115

Steady-state one-dimensional pseudo-homogeneous model

Continuity equation (negligible radial

diffusion, and constant Da):

It is one-dimensional second order ordinary

differential equation. It can be easily solved by

splitting the second order equation to two first

order ordinary differential equations.


116


117


Continuity equation (negligible radial and axial diffusions):


It is one-

dimensional first

order ordinary

differential

equation.

118

Formulation of the previous equation in terms of fractional conversion (XA)

For the derivations, see class notes.

0

2

4

)(

A

ABiA

F

rD

dz

dX

G is mass velocity and

has SI unit of kg/s·m2.

119

Significance of dimensionless numbers

It is convenient to write the above mentioned

differential equations and their boundary

conditions in terms of dimensionless groups.

Dimensionless groups may have smaller

numerical values which may facilitate a

numerical solution. Also analysis of the system

may be described in a more convenient manner.

The above equations are usually written in terms

of Peclet number.

120

Significance of dimensionless numbers

Peclet number for radial mass transfer

r

p

r

pmr

D

dG

D

duPe

,

diffusionbyratetransferM

convectionbyratetrasnferMassnumberPeclet

ass

Particle (catalyst) diameter

121

Solution of partial differential equations

• Finite difference approach

• Finite element approach

Finite difference:

• Explicit methods

(Easy to put but beware of convergence and stability)

• Implicit methods

The Crank-Nicolson method

Finite element:

• Comsol Multiphysics (old name Femlab based on MATLAB

PDE Toolbox), Fluent etc.

PDE Toolbox

http://www.google.com.pk/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=AuyukYtpk9u1zM&tbnid=Cr4_q6Zko-ExkM:&ved=0CAUQjRw&url=http%3A%2F%2Fazeotropy.com%2FCOMSOL%2F&ei=jK6uUua-EYbR0QXovICACQ&bvm=bv.57967247,d.d2k&psig=AFQjCNGQARbTLbHNQu7cmkAF3ks2zqpKqg&ust=1387266042929241

122

Workshop MATLAB PDE Toolbox

123

Polymath software

Polymath software is an excellent source for solving single and

simultaneous ordinary differential equations.

Instead of using POLYMATH software, you can prepare

an excel spreadsheet using Euler’s method or Runge-

Kutta method

124

Euler’s method

Euler’s method is employed for approximate solution of ordinary

differential equations with initial value problems.

Take interval “h” as low as possible.

Take at least 100 steps for the Euler’s method for a better accuracy.

Runge-Kutta 4th order method has better efficiency than Euler’s

method. You can easily setup these methods in Excel or

MATLAB.

Euler (1707-1783)

),(1 nnnn yxfhyy

nn xxh 1

125

4th Order RK method in MATLAB

126

Weight of catalyst in a fixed bed reactor (Use of Polymath)

A fixed bed reactor is to be designed for the

dehydrogenation of methylcyclohexane to toluene to obtain

90% conversion of the reactant. The feed is pure MCH and

reaction is clean with no side reactions.

Initial molar flowrate of MCH = 100 mol/s

Reactor pressure = 1.013 bar

Reactor temperature = 661.8 K (isothermal conditions)

Density of the catalyst bed = 860 kg/m3

, mol/s.g-cat (n = 1)

Where, k = 1.65⨯10‒5 mol/s.g-cat-Pa

yA =

127

Mathematical modeling of a liquid-liquid extractor

Acetic acid (A) is to be extracted from an organic solution

(say in ethylacetate) using a suitable solvent such as water.

The extraction is to be carried out in a differential extractor

such as liquid pulsed extraction column (installed in our unit

operations laboratory) as shown on the next slide. Apply

shell mass balance and develop the differential equations

applied to the extractor. The light phase, i.e., organic phase

which is behaving as dispersed phase enters from the bottom

while heavy phase (continuous phase) which is aqueous

phase enters from the top and flows towards bottom.

Consider isothermal conditions and negligible pressure drop.

128

Liquid pulsed extraction column [11]

Installed at unit operations

laboratory

129

Mathematical modeling of a differential extraction column

130


For dispersed phase moving from bottom

and losing “A”

For continuous phase coming from top and

gaining “A”


131


What are the boundary conditions?

Hint: Go to

Smoot, L.D. and Babb, A.L., 1962. Mass transfer studies in a pulsed extraction column:

longitudinal concentration profiles. Ind. Eng. Chem. Fundam. 1, 93‒103.

132

Homework Problem

133

Homework Problem

134

Homework Problem

135

Differential equation for mass transfer

A control volume in a body of a fluid is shown

below:

136

Differential equation for mass transfer

For the control volume shown

The net rate of mass transfer through the control

volume may be worked out by considering mass

to be transferred across the control surfaces.



137

Differential equations for mass transfer in terms of NA (molar units) [4]

∂

In rectangular coordinates


138

Differential equation for mass transfer in rectangular coordinates (mass units)

For the control volume shown before

The net rate of mass transfer through the control

volume may be worked out by considering mass

to be transferred across the control surfaces.

AAzAyAxA rz

n

y

n

x

n

t

139

Differential equations for mass transfer in terms of jAz [1]

140

Differential equations for mass transfer in terms of ωA [1]

141

Class activity: Application of general mass balance equation

142


143


144


145


146


147

Application of general mass balance equation

Apply the general continuity equations for various other

systems described in the class.

Think other situations such as falling film problem

discussed in Chapter 2 of the text book [1] and develop

the model differential equations.

148

Mathematical modeling of a differential gas-absorber

Ammonia gas (A) is to be absorbed from an air-ammonia

mixture using a suitable solvent, say, water. The gas

absorption is to be carried out in a differential absorber such

as packed column (installed in our unit operations

laboratory) as shown on the next slide. Apply shell mass

balance and develop the differential equations (for all

possible conditions including and excluding diffusivities)

applied to the gas absorber. The gas phase enters from the

bottom and flows upwards and vice versa.

149

Packed column [11]

150

References [1] Bird, R.B. Stewart, W.E. Lightfoot, E.N. 2002. Transport Phenomena. 2nd ed. John

Wiley & Sons, Inc. Singapore.

[2] Geankoplis, C.J. 1999. Transport processes and unit operations. 3rd ed. Prentice-Hall

International, Inc.

[3] Foust, A.S.; Clump, C.W.; Andersen, L.B.; Maus, L.; Wenzel, L.A. 1980. Principles

of Unit Operations. 2nd ed. John Wiley & Sons, Inc. New York.

[4] Welty, J.R.; Wicks, C.E.; Wilson, R.E.; Rorrer, G.L. 2007. Fundamentals of

momentum, heat and mass transfer. 5th ed. John Wiley & Sons, Inc.

[5] Poling, B.E.; Prausnitz, J.H.; O’Connell, J.P. 2000. The properties of gases and

liquids. 5th ed. McGraw-Hill.

[6] Lide, D.R. 2007. CRC Handbook of Chemistry and Physics. 87th ed. CRC Press.

[7] Koretsky, M.D. 2013. Engineering and chemical thermodynamics. 2nd ed. John Wiley

& Sons, Inc.

[8] Cengel, Y.A. (2003). Heat transfer: A practical approach. 2nd ed. McGraw-Hill.

[9] Staff of research and education association. The transport phenomena problem solver:

momentum, energy, mass. Research and Education Association.

[10] Missen, R.W., Mims, C.A., and Saville, B.A. 1999. Introduction to chemical reaction

engineering and kinetics. John Wiley & Sons, Inc., New York.

[11] Usman, M.R. 2015. Comprehensive dictionary of chemical engineering. Lulu

Publishing.

Transport phenomena-Mass Transfer 31-Jul-2016

Engineering

Transcript of Transport phenomena-Mass Transfer 31-Jul-2016